What are the rest energy, the kinetic energy, and the total energy of a 1.0 g particle with a speed of 0.80c?

Verified step by step guidance

1. Step 1: Start by calculating the rest energy of the particle using Einstein's equation for rest energy: $E\_r\; =\; m\; c^2$, where $m$ is the mass of the particle (1.0 g = 0.001 kg) and $c$ is the speed of light (approximately $3.0\; imes\; 10^8\; ext\{\; m/s\}$).
2. Step 2: Calculate the relativistic factor $ext\{\gamma \}$ using the formula , where $v$ is the speed of the particle (0.80c). Substitute the values to find $ext\{\gamma \}$.
3. Step 3: Use the relativistic total energy formula $E\_t\; =\; ext\{\gamma \}\; m\; c^2$ to calculate the total energy of the particle. Substitute the values of $ext\{\gamma \}$, $m$, and $c$.
4. Step 4: Determine the kinetic energy of the particle using the relationship $K\; =\; E\_t\; -\; E\_r$, where $E\_t$ is the total energy and $E\_r$ is the rest energy. Substitute the values to find $K$.
5. Step 5: Summarize the results: the rest energy, kinetic energy, and total energy of the particle are now determined. Ensure all units are consistent (e.g., joules) and verify the calculations for accuracy.

Key Concepts

Rest Energy

Rest energy is the energy contained in an object due to its mass when it is at rest, described by Einstein's famous equation E=mc². For a particle with mass m, its rest energy is calculated by multiplying its mass by the square of the speed of light (c). In this case, for a 1.0 g particle, the rest energy can be determined by converting grams to kilograms and applying the equation.

Kinetic Energy in Relativity

In the context of special relativity, the kinetic energy of an object moving at a significant fraction of the speed of light (c) is given by the formula KE = (γ - 1)mc², where γ (gamma) is the Lorentz factor. The Lorentz factor accounts for the effects of relativistic speeds and is calculated as γ = 1 / √(1 - v²/c²). For a particle moving at 0.80c, this factor must be computed to find the correct kinetic energy.

Total Energy

The total energy of a particle in relativistic physics is the sum of its rest energy and kinetic energy. It can be expressed as E_total = mc² + KE. This total energy reflects both the inherent energy due to mass and the energy due to motion, providing a complete picture of the particle's energy state at relativistic speeds.